On the transcendence of certain real numbers

In this article we discuss the transcendence of certain infinite sums and products by using the Subspace theorem. In particular we improve the result of Han\v{c}l and Rucki \cite{hancl3}.Comment: 14 page

Linear independence of values of the qq-exponential and related functions

In this paper, we derive results concerning the linear independence of values of the $q$-exponential and certain cognate functions at algebraic arguments when the associated modulus $q$ is a Pisot-Vijayraghavan number. As a consequence, we deduce the irrationality of special values of these functions, some of which were known earlier using alternate methods

On inhomogeneous extension of Thue-Roth's type inequality with moving targets

Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta, \beta\in\overline{\mathbb Q}^\times$ be algebraic numbers with $\beta$ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\in\Gamma\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $$0<|\delta qu+\beta-p|<\frac{1}{H^\varepsilon(u)q^{d+\varepsilon}},$$ where $H(u)$ denotes the absolute Weil height. As an application of this result, we also prove a transcendence result, which states as follows: Let $\alpha>1$ be a real number. Let $\beta$ be an algebraic irrational and $\lambda$ be a non-zero real algebraic number. For a given real number $\varepsilon >0$, if there are infinitely many natural numbers $n$ for which $||\lambda\alpha^n+\beta|| < 2^{- \varepsilon n}$ holds true, then $\alpha$ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha$ and $\beta$ both are algebraic satisfying same conditions, then a particular result of Kulkarni, Mavraki and Nguyen, proved in [3] asserts that $\alpha^d$ is a Pisot number. When $\beta$ is algebraic irrational, our result implies that no algebraic number $\alpha$ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [6]. The proof of our results uses the Subspace Theorem based on the idea of Corvaja and Zannier [2] together with various modification play a crucial role in the proof.Comment: To appear in International Mathematics Research Notices(IMRN

On the trace of linear combination of powers of algebraic numbers

In this article, we prove three main results. Let $\lambda_1, \ldots, \lambda_k$ and $\alpha_1, \ldots, \alpha_k$ be nonzero algebraic numbers for some integer $k\geq 1$ and let $L = \mathbb{Q}(\lambda_1, \ldots, \lambda_k, \alpha_1, \ldots, \alpha_k)$ be the number field. Let $K$ be the Galois closure of $L$ and let $h$ be the order of the torsion subgroup of $K^\times$. We first prove an extension of a result of B. de Smit [3] as follows: {\it Take $\lambda_i = b_i \in\mathbb{Q}$ for $i=1,\ldots, k$ such that $b_1+\cdots+b_k = n\ne 0$ and $\alpha_j$'s are some of the Galois conjugates (not necessarily distinct) of $\alpha_1$ for all $j = 2, \ldots, k$ and $d\geq 1$ is the degree of $\alpha_1$. If $\displaystyle{\mathrm{Tr}}_{L/\mathbb{Q}}(b_1\alpha_1^j+\cdots+b_k\alpha_k^j) \in \mathbb{Z}$ for all $j = 1, 2, \ldots, d+d[\log_2(nd)]+1$, then $\alpha_1$ is an algebraic integer.} We then prove a general result for the infinite version as follows. {\it Suppose ${\mathrm{Tr}}_{L/\mathbb{Q}}(\lambda_i\alpha_i^a) \ne 0$ for all integers $a\in \{0, 1, 2, \ldots, h-1\}$ and for all integers $i=1,\ldots,k$. If ${\mathrm{Tr}}_{L/\mathbb{Q}}(\lambda_1\alpha_1^n+\cdots +\lambda_k \alpha^n_k)\in\mathbb{Z}$ holds true for infinitely many natural numbers $n$, then each $\alpha_i$ is an algebraic integer for all $i=1,\ldots,k$. } Here the extra assumption is a necessary condition for $k > 1$ (See Remark 1.3). We also prove a Diophantine result which states: {\it For a given rational number $p/q$, there are at most finitely many natural numbers $n$ such that ${\mathrm{Tr}}_{L/\mathbb{Q}}(\lambda_1\alpha_1^n+\cdots +\lambda_k \alpha^n_k) = p/q$.

Set Equidistribution of subsets of (Z/nZ) *

In 2010, Murty and Thangadurai [MuTh10] provided a criterion for the set equidistribution of residue classes of subgroups in (Z/nZ) *. In this article, using similar methods, we study set equidistribution for some class of subsets of (Z/nZ) *. In particular, we study the set equidistribution modulo 1 of cosets, complement of subgroups of the cyclic group (Z/nZ) * and the subset of elements of fixed order, whenever the size of the subset is sufficiently large